An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

Three-dimensional water entry of a solid body: A particle image velocimetry study

Understanding and predicting the hydrodynamic loading experienced by a solid body during water impact is critical for researchers and practitioners in naval engineering. While two-dimensional (2D) water entry problems have been extensively investigated, experimental data on 3D fluid–structure interactions during water impact are rather limited. Here, particle image velocimetry (PIV) is utilized to study the free fall vertical impact of a solid body, modeling a ship hull, on an otherwise quiescent fluid. Planar PIV is used to measure the velocity field on multiple cross-sections along the length and width of the model. These data are combined to infer the 3D velocity field in the entire fluid. The 3D velocity field is then utilized to reconstruct the pressure field by integrating the incompressible 3D Navier–Stokes equations in a time-varying domain, where both the free surface and the fluid–solid interface evolve in time. By evaluating the pressure field on the wetted surface of the model, we estimate the hydrodynamic loading during water entry. Experimental results demonstrate the central role of 3D effects on both the flow physics and the hydrodynamic loading. As the cross-sectional velocity decreases away from the mid-span, we observe a robust increase in the axial velocity component. This translates into a complex spatio-temporal dependence of the hydrodynamic loading, which is initially maximized in the vicinity of the pile-up and later increases toward the keel. Due to the deceleration of the model during the impact and the increase in the wetted surface, the hydrodynamic loading close to the mid-span in the early stage of the impact is considerably larger than the ends. The 3D flow physics is used to study the energy imparted to the fluid during the impact, which we find to be mostly transferred to the risen water, consisting of the pile-up region and the spray jet. Our methodology can be implemented for the analysis of other solid bodies with multiple geometric curvatures, and our experimental results can be utilized for the validation of 3D mathematical models of water entry.     

Numerical simulations of wave interactions with vertical wave barriers using the SPH method

This paper focuses on the fluid boundary separation problem of the conventional dynamic solid boundary treatment (DSBT) and proposes a modified DSBT (MDSBT). Classic 2D free dam break flows and 3D dam break flows against a rectangular box are used to assess the performance of this MDSBT in free surface flow and violent fluid–structure interaction, respectively. Another test, water column oscillations in a U‐tube, is specially designed to reveal the applicability of dealing with two types of particular boundaries: the wet–dry solid boundary and the large‐curvature solid boundary. A comparison between the numerical results and the experimental data shows that the MDSBT is capable of eliminating the fluid boundary separation, improving the accuracy of the solid boundary pressure calculations and preventing the unphysical penetration of fluid particles. Using a 2D SPH numerical wave tank with MDSBT, the interactions between regular waves and a simplified vertical wave barrier are simulated. The numerical results reveal that the maximum horizontal force occurs at the endpoint of the vertical board, and with the enlargement of the relative submerged board length, the maximum moment grows linearly; furthermore, the relative average mass transportation under the breakwater initially increases to 11.14 per wave strike but is later reduced. The numerical simulation of a full‐scale 3D wave barrier with two vertical boards shows that the wave and structure interactions in the practical project are far more complicated than in the simplified 2D models. The SPH model using the MDSBT is capable of providing a reference for engineering designs. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear water wave interaction with floating bodies in SPH

A weakly compressible SPH solver is presented for applications involving nonlinear interaction between water waves and floating bodies. A complete algorithm able to compute fully coupled viscous Fluid–Solid interactions is described. No slip boundary condition on the solid surface is enforced through a ghost–fluid technique and the global loads are evaluated through the momentum exchange between fluid and ghost particles. A dedicated algorithm is developed to manage the intersection between the free surface and the solid profile. An explicit synchronous algorithm is proposed for the full coupling between fluid and rigid bodies. Stability, convergence and conservation properties are tested on several freely floating test cases and a final validation of the full algorithm is performed for the interaction between a 2-D box and a wave packet.     

A corrected symmetric SPH method to simulate viscoelastic free surface flows based on the PTT model

In this work, a corrected symmetric and periodic density reinitialized SPH (CSPDR‐SPH) method is proposed and extended to simulate the viscoelastic free surface flows based on the Phan–Thien–Tanner model. The improvements mainly lie in deriving a corrected symmetric kernel gradient, and combining it with a periodic density reinitialization procedure. In addition, a simple artificial viscosity and a simple artificial stress form are adopted. Thus, the CSPDR‐SPH method has higher accuracy and better stability than the SPH method, and conserves both linear and angular momentums. The consistency and convergence of the CSPDR‐SPH method are justified by approximating a function in one and two dimensions. The merits of CSPDR‐SPH method are demonstrated by several benchmarks. The simple flow in a two‐dimensional channel is investigated to show the capability of the CSPDR‐SPH method to simulate the viscoelastic free surface flow. Then the CSPDR‐SPH method is extended to simulate the impacting drop problem. Numerical results show that the CSPDR‐SPH method can precisely capture the viscoelastic free surface. The Reynolds number, Weissenberg number and elongation parameter have remarkable effect on the flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

A comparative study of truly incompressible and weakly compressible SPH methods for free surface incompressible flows

In this paper, the performance of the incompressible SPH (ISPH) method and an improved weakly compressible SPH (IWCSPH) method for free surface incompressible flows are compared and analyzed. In both methods, the Navier–Stokes equations are solved, and no artificial viscosity is used. The ISPH algorithm in this paper is based on the classical SPH projection method with common treatments on solid boundaries and free surfaces. The IWCSPH model includes some advanced corrective algorithms in density approximation and solid boundary treatment (SBT). In density approximation, the moving least squares (MLS) approach is applied to re‐initialize density every several steps to obtain smoother and more stable pressure fields. An improved coupled dynamic SBT algorithm is implemented to obtain stable pressure values near solid wall areas and, thus, to minimize possible numerical oscillations brought in by the solid boundaries. Three representative numerical examples, including a benchmark test for hydrostatic pressure, a dam breaking problem and a liquid sloshing problem, are comparatively analyzed with ISPH and IWCSPH. It is demonstrated that the present IWCSPH is more attractive than ISPH in modeling free surface incompressible flows as it is more accurate and more stable with comparable or even less computational efforts. Copyright © 2013 John Wiley & Sons, Ltd.     

A critical investigation of smoothed particle hydrodynamics applied to problems with free‐surfaces

In this paper, an in‐depth study of SPH method, in its original weakly compressible version, is achieved on dedicated 2D and 3D free‐surface flow test cases. These rather critical prototype problems shall constitute suitable test cases to get through when building a free‐surface SPH model. The present work aims at investigating various numerical aspects of this method, often little mentioned in literature. In particular, a great care is paid to the dynamic part of the solution, which is critical to the local hydrodynamic load prediction. The role of numerical errors in the development of acoustic frequencies in the pressure signals is discussed, as well as the influence of the choice of the sound velocity. On the shown test problems, it is also evidenced that some numerical tools are crucial to ensure the robustness and accuracy of the standard SPH method. The convergence of our model is heuristically proved on these nonlinear prototype tests, showing at the same time the very satisfactory level of accuracy reached. Through these tests, some other numerical specificities of the SPH method are discussed, such as the self‐redistribution of the particles occurring during the Lagrangian evolution. A higher order model is also proposed, and its advantages and drawbacks are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

The capturing of free surfaces in incompressible multi‐fluid flows

By treating it as a contact discontinuity in the density field, a free surface between two immiscible fluids can be automatically ‘captured’ by the enforcement of conservation laws. A surface‐capturing method of this kind requires no special tracking or fitting treatment for the free surface, thereby offering the advantage of algorithm simplicity over the surface‐tracking or the surface‐fitting method. A surface‐capturing method based on a new multi‐fluid incompressible Navier–Stokes formulation is developed. It is applied to a variety of free‐surface flows, including the Rayleigh–Taylor instability problem, the ship waves around a Wigley hull and a model bubble‐rising problem to demonstrate the validity and versatility of the present method. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical simulation of landslide impulsive waves by incompressible smoothed particle hydrodynamics

An incompressible‐smoothed particle hydrodynamics (I‐SPH) formulation is presented to simulate impulsive waves generated by landslides. The governing equations, Navier–Stokes equations, are solved in a Lagrangian form using a two‐step fractional method. Landslides in this paper are simulated by a submerged mass sliding along an inclined plane. During sliding, both rigid and deformable landslides mass are considered. The present numerical method is examined for a rigid wedge sliding into water along an inclined plane. In addition solitary wave generated by a heavy box falling inside water, known as Scott Russell wave generator, which is an example for simulating falling rock avalanche into artificial and natural reservoirs, is simulated and compared with experimental results. The numerical model is also validated for gravel mass sliding along an inclined plane. The sliding mass approximately behaves like a non‐Newtonian fluid. A rheological model, implemented as a combination of the Bingham and the general Cross models, is utilized for simulation of the landslide behaviour. In order to match the experimental data with the computed wave profiles generated by deformable landslides, parameters of the rheological model are adjusted and the numerical model results effectively match the experimental results. The results prove the efficiency and applicability of the I‐SPH method for simulation of these kinds of complex free surface problems. Copyright © 2007 John Wiley & Sons, Ltd.     

Global volume conservation in unsteady free surface flows with energy absorbing far‐end boundaries

A wave absorption filter for the far‐end boundary of semi‐infinite large reservoirs is developed for numerical simulation of unsteady free surface flows. Mathematical model is based on finite volume solution of the Navier–Stokes equations and depth‐integrated continuity equation to track the free surface. The Sommerfeld boundary condition is applied at the far‐end of the truncated computational domain. A dissipation zone is formed by applying artificial pressure on water surface to dissipate the kinetic energy of the outgoing waves. The computational scheme is tested to verify the conservation of total fluid volume in the domain for long simulation durations. Combination of the Sommerfeld boundary and dissipation zone can effectively minimize reflections and prevent cumulative changes in total fluid volume in the domain. Solitary wave, nonlinear periodic waves and irregular waves are simulated to illustrate the numerical developments. Earthquake excited surface waves and nonlinear hydrodynamic pressures in a dam–reservoir are computed. Copyright © 2009 John Wiley & Sons, Ltd.     

Quadratic programming algorithm for wall slip and free boundary pressure condition

Wall slip is often observed in a highly sheared fluid film in a solid gap. This makes a difficulty in mathematical analysis for the hydrodynamic effect because fluid velocity at the liquid–solid interfaces is not known a priori. If the gap has a convergent–divergent wedge, a free boundary pressure condition, i.e. Reynolds pressure boundary condition, is usually used in the outlet zone in numerical solution. This paper, based on finite element method and parametric quadratic programming technique, gives a numerical solution technique for a coupled boundary non‐linearity of wall slip and free boundary pressure condition. It is found that the numerical error decreases with the number of elements in a negative power law having an index larger than 2. Our method does not need an iterative process and can simultaneously gives rise to fluid film pressure distribution, wall slip velocity and surface shear stress. Wall slip always decreases the hydrodynamic pressure. Large wall slip even causes a null hydrodynamic pressure in a pure sliding solid gap. Copyright © 2005 John Wiley & Sons, Ltd.     

A fully hydrodynamic model for three‐dimensional,free‐surface flows

The hydrostatic pressure assumption has been widely used in studying water movements in rivers, lakes, estuaries, and oceans. While this assumption is valid in many cases and has been successfully used in numerous studies, there are many cases where this assumption is questionable. This paper presents a three‐dimensional, hydrodynamic model for free‐surface flows without using the hydrostatic pressure assumption. The model includes two predictor–corrector steps. In the first predictor–corrector step, the model uses hydrostatic pressure at the previous time step as an initial estimate of the total pressure field at the new time step. Based on the estimated pressure field, an intermediate velocity field is calculated, which is then corrected by adding the non‐hydrostatic component of the pressure to the estimated pressure field. A Poisson equation for non‐hydrostatic pressure is solved before the second intermediate velocity field is calculated. The final velocity field is found after the free surface at the new time step is computed by solving a free‐surface correction equation. The numerical method was validated with several analytical solutions and laboratory experiments. Model results agree reasonably well with analytical solutions and laboratory results. Model simulations suggest that the numerical method presented is suitable for fully hydrodynamic simulations of three‐dimensional, free‐surface flows. Copyright © 2003 John Wiley & Sons, Ltd.     

Superquadric DEM-SPH coupling method for interaction between non-spherical granular materials and fluids

The research on the coupling method of non-spherical granular materials and fluids aims to predict the particle–fluid interaction in this study. A coupling method based on superquadric elements is developed to describe the interaction between non-spherical solid particles and fluids. The discrete element method (DEM) and the smoothed particle hydrodynamics (SPH) are adopted to simulate granular materials and fluids. The repulsive force model is adopted to calculate the coupling force and then a contact detection method is established for the interaction between the superquadric element and the fluid particle. The contact detection method captures the shape of superquadric element and calculates the distance from the fluid particle to the surface of superquadric element. Simulation cases focusing on the coupling force model, energy transfer, and large-scale calculations have been implemented to verify the validity of the proposed coupling method. The coupling force model accurately represents the water entry process of a spherical solid particle, and reasonably reflects the difference of solid particles with different shapes. In the water entry process of multiple solid particles, the total energy of the water entry process of multiple solid particles tends to be stable. The collapse process of the partially submerged granular column is simulated and analyzed under different parameters. Therefore, this coupling method is suitable to simulate fluid–particle systems containing solid particles with multiple shapes.     

Conservation of circulation in SPH for 2D free‐surface flows

In this paper, we study how accurately the Smoothed Particle Hydrodynamics (SPH) scheme accounts for the conservation and the generation of vorticity and circulation, in a low viscosity, weakly compressible, barotropic fluid in the context of free‐surface flows. We consider a number of simple examples to clarify the processes involved and the accuracy of the simulations. The first example is a differentially rotating fluid where the integration path for the circulation becomes progressively more complicated, whereas the structure of the velocity field remains simple. The second example is the collision of two rectangular regions of fluid. We show that SPH accurately predicts the time variation of the circulation as well as the total vorticity for selected domains advected by the fluid. Finally, a breaking wave is considered. For such a problem we show how the dynamics of the vorticity generated by the breaking process is captured by the SPH model. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

A detailed study of lid‐driven cavity flow at moderate Reynolds numbers using Incompressible SPH

Lid‐driven cavity flow at moderate Reynolds numbers is studied here, employing a mesh‐free method known as Smoothed Particle Hydrodynamics (SPH). In a detailed study of this benchmark, the incompressible SPH approach is applied together with a particle shifting algorithm. Additionally, a new treatment for no‐slip boundary conditions is developed and tested. The use of the aforementioned numerical treatment for solid walls leads to significant improvements in the results with respect to other SPH simulations carried out with similar spatial resolution. However, the effect of spatial resolution is not considered in the present study as the number of particles used in each case was kept constant, approximately reproducing the same resolutions employed in reference studies available in the literature as well. Altogether, the detailed comparisons of field variables at discreet points demonstrate the accuracy and robustness of the new SPH method. Copyright © 2014 John Wiley & Sons, Ltd.     

An SPH pressure correction algorithm for multiphase flows with large density ratio

In this paper, we propose an interfacial pressure correction algorithm for smoothed particle hydrodynamics (SPH) simulation of multiphase flows with large density ratios. This correction term is based on the assumption of small deformation of the interface, and derived from perturbation expansion analysis. It is also proven to be applicable in cases with complex interfaces. This correction algorithm helps to overcome the discontinuities of the pressure gradient over the interfaces, which may cause unphysical gap between different phases. This proposed correction algorithm is implemented on a recent multiphase SPH model, which is based on the assumption of pressure continuity over the interfaces. The coupled dynamic solid boundary treatment is used to simulate solid walls; and a cut‐off pressure is applied to avoid negative particle pressure, which may cause computational instabilities in SPH. Three numerical examples of air–water flows, including sloshing, dam breaking, and water entry, are presented and compared with experimental data, indicating the robustness of our pressure correction algorithm in multiphase simulations with large density ratios. Copyright © 2015 John Wiley & Sons, Ltd.     

A meshless method for numerical simulation of depth‐averaged turbulence flows using a k‐ϵ model

A three‐dimensional numerical model is developed to analyze free surface flows and water impact problems. The flow of an incompressible viscous fluid is solved using the unsteady Navier–Stokes equations. Pseudo‐time derivatives are introduced into the equations to improve computational efficiency. The interface between the two phases is tracked using a volume‐of‐fluid interface tracking algorithm developed in a generalized curvilinear coordinate system. The accuracy of the volume‐of‐fluid method is first evaluated by the multiple numerical benchmark tests, including two‐dimensional and three‐dimensional deformation cases on curvilinear grids. The performance and capability of the numerical model for water impact problems are demonstrated by simulations of water entries of the free‐falling hemisphere and cone, based on comparisons of water impact loadings, velocities, and penetrations of the body with experimental data. For further validation, computations of the dam‐break flows are presented, based on an analysis of the wave front propagation, water level, and the dynamic pressure impact of the waves on the downstream walls, on a specific container, and on a tall structure. Extensive comparisons between the obtained solutions, the experimental data, and the results of other numerical simulations in the literature are presented and show a good agreement. Copyright © 2015 John Wiley & Sons, Ltd.